On Diamond’s L1 Criterion for Asymptotic Density of Beurling Generalized Integers

Abstract

We give a short proof of the $L^1$ criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)={\sum }_{n_k\le x}\mu (n_k)/n_k=o(1)$ with the Beurling analog $\mu$ of the Möbius function.